From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 20 Oct 2022 18:25:31 +0200 Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex' --- source/know/concept/ito-integral/index.md | 107 +++++++++++++++--------------- 1 file changed, 54 insertions(+), 53 deletions(-) (limited to 'source/know/concept/ito-integral') diff --git a/source/know/concept/ito-integral/index.md b/source/know/concept/ito-integral/index.md index 3f17a9a..f087f97 100644 --- a/source/know/concept/ito-integral/index.md +++ b/source/know/concept/ito-integral/index.md @@ -9,10 +9,10 @@ layout: "concept" --- The **Itō integral** offers a way to integrate -a given [stochastic process](/know/concept/stochastic-process/) $G_t$ -with respect to a [Wiener process](/know/concept/wiener-process/) $B_t$, +a given [stochastic process](/know/concept/stochastic-process/) $$G_t$$ +with respect to a [Wiener process](/know/concept/wiener-process/) $$B_t$$, which is also a stochastic process. -The Itō integral $I_t$ of $G_t$ is defined as follows: +The Itō integral $$I_t$$ of $$G_t$$ is defined as follows: $$\begin{aligned} \boxed{ @@ -22,17 +22,17 @@ $$\begin{aligned} } \end{aligned}$$ -Where have partitioned the time interval $[a, b]$ into steps of size $h$. -The above integral exists if $G_t$ and $B_t$ are adapted -to a common filtration $\mathcal{F}_t$, -and $\mathbf{E}[G_t^2]$ is integrable for $t \in [a, b]$. -If $I_t$ exists, $G_t$ is said to be **Itō-integrable** with respect to $B_t$. +Where have partitioned the time interval $$[a, b]$$ into steps of size $$h$$. +The above integral exists if $$G_t$$ and $$B_t$$ are adapted +to a common filtration $$\mathcal{F}_t$$, +and $$\mathbf{E}[G_t^2]$$ is integrable for $$t \in [a, b]$$. +If $$I_t$$ exists, $$G_t$$ is said to be **Itō-integrable** with respect to $$B_t$$. ## Motivation -Consider the following simple first-order differential equation for $X_t$, -for some function $f$: +Consider the following simple first-order differential equation for $$X_t$$, +for some function $$f$$: $$\begin{aligned} \dv{X_t}{t} @@ -40,7 +40,7 @@ $$\begin{aligned} \end{aligned}$$ This can be solved numerically using the explicit Euler scheme -by discretizing it with step size $h$, +by discretizing it with step size $$h$$, which can be applied recursively, leading to: $$\begin{aligned} @@ -51,7 +51,7 @@ $$\begin{aligned} \approx X_0 + \sum_{s = 0}^{s = t} f(X_s) \: h \end{aligned}$$ -In the limit $h \to 0$, this leads to the following unsurprising integral for $X_t$: +In the limit $$h \to 0$$, this leads to the following unsurprising integral for $$X_t$$: $$\begin{aligned} \int_0^t f(X_s) \dd{s} @@ -59,18 +59,18 @@ $$\begin{aligned} \end{aligned}$$ In contrast, consider the *stochastic differential equation* below, -where $\xi_t$ represents white noise, -which is informally the $t$-derivative -of the Wiener process $\xi_t = \idv{B_t}{t}$: +where $$\xi_t$$ represents white noise, +which is informally the $$t$$-derivative +of the Wiener process $$\xi_t = \idv{B_t}{t}$$: $$\begin{aligned} \dv{X_t}{t} = g(X_t) \: \xi_t \end{aligned}$$ -Now $X_t$ is not deterministic, -since $\xi_t$ is derived from a random variable $B_t$. -If $g = 1$, we expect $X_t = X_0 + B_t$. +Now $$X_t$$ is not deterministic, +since $$\xi_t$$ is derived from a random variable $$B_t$$. +If $$g = 1$$, we expect $$X_t = X_0 + B_t$$. With this in mind, we introduce the **Euler-Maruyama scheme**: $$\begin{aligned} @@ -80,7 +80,7 @@ $$\begin{aligned} &= X_t + g(X_t) \: (B_{t+h} - B_t) \end{aligned}$$ -We would like to turn this into an integral for $X_t$, as we did above. +We would like to turn this into an integral for $$X_t$$, as we did above. Therefore, we state: $$\begin{aligned} @@ -89,8 +89,8 @@ $$\begin{aligned} \end{aligned}$$ This integral is *defined* as below, -analogously to the first, but with $h$ replaced by -the increment $B_{t+h} \!-\! B_t$ of a Wiener process. +analogously to the first, but with $$h$$ replaced by +the increment $$B_{t+h} \!-\! B_t$$ of a Wiener process. This is an Itō integral: $$\begin{aligned} @@ -104,14 +104,14 @@ see the [Itō calculus](/know/concept/ito-calculus/). ## Properties -Since $G_t$ and $B_t$ must be known (i.e. $\mathcal{F}_t$-adapted) -in order to evaluate the Itō integral $I_t$ at any given $t$, -it logically follows that $I_t$ is also $\mathcal{F}_t$-adapted. +Since $$G_t$$ and $$B_t$$ must be known (i.e. $$\mathcal{F}_t$$-adapted) +in order to evaluate the Itō integral $$I_t$$ at any given $$t$$, +it logically follows that $$I_t$$ is also $$\mathcal{F}_t$$-adapted. Because the Itō integral is defined as the limit of a sum of linear terms, it inherits this linearity. -Consider two Itō-integrable processes $G_t$ and $H_t$, -and two constants $v, w \in \mathbb{R}$: +Consider two Itō-integrable processes $$G_t$$ and $$H_t$$, +and two constants $$v, w \in \mathbb{R}$$: $$\begin{aligned} \int_a^b v G_t + w H_t \dd{B_t} @@ -119,7 +119,7 @@ $$\begin{aligned} \end{aligned}$$ By adding multiple summations, -the Itō integral clearly satisfies, for $a < b < c$: +the Itō integral clearly satisfies, for $$a < b < c$$: $$\begin{aligned} \int_a^c G_t \dd{B_t} @@ -127,8 +127,8 @@ $$\begin{aligned} \end{aligned}$$ A more interesting property is the **Itō isometry**, -which expresses the expectation of the square of an Itō integral of $G_t$ -as a simpler "ordinary" integral of the expectation of $G_t^2$ +which expresses the expectation of the square of an Itō integral of $$G_t$$ +as a simpler "ordinary" integral of the expectation of $$G_t^2$$ (which exists by the definition of Itō-integrability): $$\begin{aligned} @@ -144,14 +144,14 @@ $$\begin{aligned} @@ -221,9 +222,9 @@ since true white noise cannot be biased. -- cgit v1.2.3